## You are here

Homesimple example of composed conformal mapping

## Primary tabs

# simple example of composed conformal mapping

Let’s consider the mapping

$f\colon\mathbb{C}\to\mathbb{C}\quad\mathrm{with}\quad f(z)=az\!+\!b,$ |

Because $f^{{\prime}}(z)\equiv a\neq 0$, the mapping is conformal in the whole $z$-plane. Denote $\displaystyle a:=\varrho e^{{i\alpha}}$ (where $\varrho,\,\alpha\in\mathbb{R}$) and

$\displaystyle z_{1}:=\varrho z,$ | (1) |

$\displaystyle z_{2}:=e^{{i\alpha}}z_{1},$ | (2) |

$\displaystyle w:=z_{2}\!+\!b.$ | (3) |

Then the mapping $z\mapsto z_{1}$ means a dilation in the complex plane, the mapping $z_{1}\mapsto z_{2}$ a rotation by the angle $\alpha$ and the mapping $z_{2}\mapsto w$ a translation determined by the vector from the origin to the point $b$. Thus $f$ is composed of these three consecutive mappings which all are conformal.

Type of Math Object:

Example

Major Section:

Reference

Parent:

## Mathematics Subject Classification

30E20*no label found*53A30

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections